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Iterative image reconstruction for CT

Jeffrey A. Fessler

EECS Dept., BME Dept., Dept. of RadiologyUniversity of Michigan

http://www.eecs.umich.edu/fessler

AAPM Image Educational Course - Image Reconstruction II

Aug. 2, 2011

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Full disclosure

Research support from GE Healthcare Research support to GE Global Research Work supported in part by NIH grant R01-HL-098686 Research support from Intel

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Credits

Current students / post-docs

Jang Hwan Cho Se Young Chun Donghwan Kim Yong Long Madison McGaffin Sathish Ramani Stephen Schmitt

GE collaborators

Jiang Hsieh Jean-Baptiste Thibault Bruno De Man

CT collaborators

Mitch Goodsitt, UM Ella Kazerooni, UM Neal Clinthorne, UM Paul Kinahan, UW

Former PhD students (who did/do CT)

Wonseok Huh, Bain & Company Hugo Shi, Enthought Joonki Noh, Emory Somesh Srivastava, JHU Rongping Zeng, FDA Yingying, Zhang-OConnor, RGM Advisors Matthew Jacobson, Xoran Sangtae Ahn, GE Idris Elbakri, CancerCare / Univ. of Manitoba Saowapak Sotthivirat, NSTDA Thailand Web Stayman, JHU Feng Yu, Univ. Bristol Mehmet Yavuz, Qualcomm Hakan Erdogan, Sabanci University

Former MS / undegraduate students

Kevin Brown, Philips Meng Wu, Stanford ...

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Statistical image reconstruction: CT revolution

A picture is worth 1000 words (and perhaps several 1000 seconds of computation?)

Thin-slice FBP ASIR Statistical

Seconds A bit longer Much longer

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Why statistical/iterative methods for CT?

Accurate physics models X-ray spectrum, beam-hardening, scatter, ...

= reduced artifacts? quantitative CT? X-ray detector spatial response, focal spot size, ...

= improved spatial resolution? detector spectral response (e.g., photon-counting detectors)

= improved contrast?

Nonstandard geometries transaxial truncation (wide patients) long-object problem in helical CT irregular sampling in next-generation geometries coarse angular sampling in image-guidance applications limited angular range (tomosynthesis) missing data, e.g., bad pixels in flat-panel systems

Appropriate models of measurement statistics weighting reduces influence of photon-starved rays (cf. FBP)

= reducing image noise or X-ray dose

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and more...

Object constraints nonnegativity object support piecewise smoothness object sparsity (e.g., angiography) sparsity in some basis motion models dynamic models ...

Disadvantages? Computation time (super computer) Must reconstruct entire FOV Model complexity Software complexity Algorithm nonlinearities Difficult to analyze resolution/noise properties (cf. FBP) Tuning parameters Challenging to characterize performance

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Iterative vs Statistical

Traditional successive substitutions iterations e.g., Joseph and Spital (JCAT, 1978) bone correction usually only one or two iterations not statistical

Algebraic reconstruction methods Given sinogram data yyy and system model AAA, reconstruct object xxx by

solving yyy = AAAxxx ART, SIRT, SART, ... iterative, but typically not statistical Iterative filtered back-projection (FBP):

xxx(n+1) = xxx(n) + stepsize

FBP( yyy

data

AAAxxx(n)forwardproject

)

Statistical reconstruction methods Image domain Sinogram domain Fully statistical (both) Hybrid methods (e.g., AIR, SPIE 7961-18, Bruder et al.)

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Statistical methods: Image domain

Denoising methods

sinogramyyy

FBP noisy

reconstructionxxx

iterativedenoiser

final

imagexxx

Relatively fast, even if iterative Remarkable advances in denoising methods in last decade

Zhu & Milanfar, T-IP, Dec. 2010, using steering kernel regression (SKR) method

Challenges: Typically assume white noise Streaks in low-dose FBP appear like edges (highly correlated noise)

Image denoising methods guided by data statistics

sinogramyyy

FBP noisy

reconstructionxxx

magicaliterativedenoiser

sinogramstatistics?

final

imagexxx

Image-domain methods are fast (thus very practical) ASIR? IRIS? ... The technical details are often a mystery...

Challenges: FBP often does not use all data efficiently (e.g., Parker weighting) Low-dose CT statistics most naturally expressed in sinogram domain

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Statistical methods: Sinogram domain

Sinogram restoration methods

noisysinogram

yyy

adaptive

or iterativedenoiser

cleaned

sinogramyyy

FBP final

imagexxx

Adaptive: J. Hsieh, Med. Phys., 1998; Kachelrie, Med. Phys., 2001, ... Iterative: P. La Riviere, IEEE T-MI, 2000, 2005, 2006, 2008 Relatively fast even if iterative

Challenges: Limited denoising without resolution loss Difficult to preserve edges in sinograms

FBP, 10 mA FBP from denoised sinogramWang et al., T-MI, Oct. 2006, using PWLS-GS on sinogram

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(True? Fully? Slow?) Statistical image reconstruction

Object model Physics/system model Statistical model Cost function (log-likelihood + regularization) Iterative algorithm for minimization

Find the image xxx that best fits the sinogram data yyy according to the physicsmodel, the statistical model and prior information about the object

ModelSystem

Iteration

Parameters

MeasurementsProjection

Calibration ...

xxx(n) xxx(n+1)

Repeatedly revisiting the sinogram data can use statistics fully Repeatedly updating the image can exploit object properties ... greatest potential dose reduction, but repetition is expensive...

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History: Statistical reconstruction for PET

Iterative method for emission tomography (Kuhl, 1963)

FBP for PET (Chesler, 1971)

Weighted least squares for 3D SPECT (Goitein, NIM, 1972)

Richardson/Lucy iteration for image restoration (1972, 1974)

Poisson likelihood (emission) (Rockmore and Macovski, TNS, 1976)

Expectation-maximization (EM) algorithm (Shepp and Vardi, TMI, 1982)

Regularized (aka Bayesian) Poisson emission reconstruction(Geman and McClure, ASA, 1985)

Ordered-subsets EM (OSEM) algorithm (Hudson and Larkin, TMI, 1994)

Commercial release of OSEM for PET scanners circa 1997

Today, most (all?) commercial PET systems include unregularized OSEM.

15 years between key EM paper (1982) and commercial adoption (1997)(25 years if you count the R/L paper in 1972 which is the same as EM)

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Key factors in PET

OS algorithm accelerated convergence by order of magnitude Computers got faster (but problem size grew too) Key clinical validation papers? Key numerical observer studies? Nuclear medicine physicians grew accustomed to appearance

of images reconstructed using statistical methods

FBP: ML-EM:

Llacer et al., 1993

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Whole-body PET example

FBP ML-OSEM

Meikle et al., 1994

Key factor in PET: modeling measurement statistics

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History: Statistical reconstruction for CT

Iterative method for X-ray CT (Hounsfield, 1968)

ART for tomography (Gordon, Bender, Herman, JTB, 1970)

...

Roughness regularized LS for tomography (Kashyap & Mittal, 1975)

Poisson likelihood (transmission) (Rockmore and Macovski, TNS, 1977)

EM algorithm for Poisson transmission (Lange and Carson, JCAT, 1984)

Iterative coordinate descent (ICD) (Sauer and Bouman, T-SP, 1993)

Ordered-subsets algorithms(Manglos et al., PMB 1995)

(Kamphuis & Beekman, T-MI, 1998)(Erdogan & Fessler, PMB, 1999)

...

Commercial introduction of ICD for CT scanners circa 2010

( numerous omissions, including the many denoising methods)

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RSNA 2010

Zhou Yu, Jean-Baptiste Thibault, Charles Bouman, Jiang Hsieh, Ken Sauer

https://engineering.purdue.edu/BME/AboutUs/News/HomepageFeatures/ResultsofPurdueResearchUnveiledatRSNA

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MBIR example: Routine chest CT

Helical chest CT study with dose = 0.09 mSv.Typical CXR effective dose is about 0.06 mSv. Source: Health Physics Society.http://www.hps.org/publicinformation/ate/q2372.html

FBP MBIR

Veo (MBIR) is 510(k) pending. Not available for sale in the U.S.

Images courtesy of Jiang Hsieh, GE Healthcare

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Five Choices for Statistical Image Reconstruction

1. Object model

2. System physical model

3. Measurement statistical model

4. Cost function: data-mismatch and regularization

5. Algorithm / initialization

No perfect choices - one can critique all approaches!

Historically these choices are often left implicit in publications,but being explicit facilitates reproducibility.

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Choice 1. Object Parameterization

Finite measurements: {yi}Mi=1. Continuous object: f (~r) = (~r).

All models are wrong but some models are useful.

Linear series expansion approach. Represent f (~r) by xxx = (x1, . . . ,xN) where

f (~r) f (~r) =N

j=1

x j b j(~r) basis functions

Reconstruction problem becomes discrete-discrete: estimate xxx from yyy

Numerous basis functions in literature. Two primary contenders: voxels blobs (Kaiser-Bessel functions)

+ Blobs are approximately band-limited (reduced aliasing?) Blobs have larger footprints, increasing computation.

Open question: how small should the voxels be?

One practical compromise: wide FOV coarse-grid reconstruction followedby fine-grid refinement over ROI, e.g., Ziegler et al., Med. Phys., Apr. 2008

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Global reconstruction: An inconvenient truth